#144693

order	512
nilpotency class	4
isoclinism	isoclinic class of #142258
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

431 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(2,3)(5,6,8,13)(7,16,18,35)(9,23,22,32)(10,15,11,27)(12,29,30,19)(14,28,24,26)(17,25,33,34)(20,31,21,36);(1,2)(3,4)(5,6)(7,17)(8,13)(9,24)(10,27)(11,15)(12,20)(14,22)(16,34)(18,33)(19,31)(21,30)(23,28)(25,35)(26,32)(29,36);(1,3,4,2)(5,7,8,18)(6,12,13,30)(9,25,22,34)(10,21,11,20)(14,31,24,36)(15,33,27,17)(16,32,35,23)(19,28,29,26)

Polycyclic presentation:

2^9|a^2=i,c^2=d,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51889);